CONFORMAL MAPPING FOR SURFACE MODELLING

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CONFORMAL MAPPING FOR SURFACE MODELLING

N. Andion, C. de Castilho

To cite this version:

N. Andion, C. de Castilho. CONFORMAL MAPPING FOR SURFACE MODELLING. Journal de

Physique Colloques, 1989, 50 (C8), pp.C8-21-C8-26. �10.1051/jphyscol:1989804�. �jpa-00229902�

COLLOQUE DE PHYSIQUE

Colloque C8, supplbment au n o 11, Tome 50, novembre 1989

CONFORMAL MAPPING FOR SURFACE MODELLING

N.P. ANDION and C.M.C. DE CASTILHO

I n s t i t u t o d e F i s i c a , UFBa., Campus d a F e d e r a ~ s o , 40 210 S a l v a d o r , B a h i a , B r a z i l

Abstract

-

A confolmal mapping method for obtaining electric potentials in a region close to a structured surface is developed. By neglecting one of the surface dimensions and assuming periodicity along the other, the resulting tm- dimensional problem can be analytically treated. A s examples of the method some arrays of monopoles and dipoles are calculated from which a surface model is proposed and exactly solved.

Many different methods exist for studying the electric field and potential close to a surface.

Some authors prefer approaches based on j e l l i m model /1,2/ and so do not take into account the crystalline structure, while others do include it /3-5/ in different m r i c a l approaches.

Confolmal mapping has been used in several vector field problems, f m hydrodynamcs t o electrostatics /6/ and even on typical Surface phenomena problems it has been applied /7/.

Following a purely analytical rTtrackTr a method is here developed f o r dealing with t w o - dimensional electrostatic problems with periodicity i n one direction. The third actual coordinate, which belongs to the surface plane, is taken as ignorable. The method is based on a conformal mapping leading to a new simplified problem with broad physical andlogy with the original one. Consistency with more r e a l i s t i c point planar arrays is brought up by an

"integral equivalencerr hypothesis. A l l quantities are expressed in S I units.

2 - MAPPING RIE EUTXXOSTATIC PROBLEM

Let @(x,y) be an electrostatic potential having period L in x. It w i l l , a t f i r s t , be worked out as a flmction of the complex variable

r

⁼ x

+

i y which defines the conplex r-plane. Let

represent a mapping onto a w-plane where w ^{= P} exp (iv). Therefore

r, = ( - = / a ) log w (2)

represents the inverse transformation. Geometrical features of this mapping are shown i n fig.

1. W c t i o n (w) is analytic except for w = 0. According to the theory of complex variables /6/, this implies that a flmction f ( c ) which is harmonic i n a certain domain of the r-plane is transformed, thrwgh eq. 2, into an harmonic g(w) i n the corresponding domain of the w- plane, provided that the values of p = 0 and, correspondingly, y tending t o infinity are excluded. Conversely, w(r) in eq. 1 is analytic thrwghout the S-plane and therefore always transforms harmonic flmctions of w into harmonic m t i o n s i n the corresponding domain of the C-plane. Furthermore these mappings are also conformal thrcughout the P- and w-pianes.

Indeed, unless for the t r i v i a l exclusion of p = 0 a+

I r I-. -,

both the conditions of being analytic and having non-zem derivatives are accomplished for the mapping k t i o n s given by eqs. 1 and 2. For dealing with quantities which are defined in three dimensions, to the real form

v = 2nx/~; p = exp (-2ny/~) (3)

of eq. 1, a convenient antisymmetrical t r a n s f o r m a t i o n i n t h e t h i r d d i r e c t i o n ,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989804

s = -2, is added with z and s being the conventionally defined third axes coordinates of cartesian (x, y, z) and cylindrical (p, v, s) systems. Conversely,

x = ~u/2n

,

y = (-~/2n) lnp

,

z = -s (4) represent three dimensional extensions of eq. 2 and its transformation. From these equations, the transfomtion jacobian is obtained: I = (L2/4 n2p).!Jhey also imply that for electrostatic potentials, with translational symnetry along third axes, laplacian operations on

c-

and w-planes are related by

v2 e

(x,y) = (2 ~ - P / L ) ~ v;F( P .V )

5 ⁽⁵⁾

where F(p

,

v) is the transform of e (x,y) according to eqs. 4. Correlating this equation with the jacobian leads to

dqc = dqw; (pfO), (6)

This means that throughthis mappirg m t only the hamonic character of the flmctions but also the charges involved in a given electrostatic problem are preserved between domains such that P f 0. For P= 0 the singularity may be solved as follows. Given a system, bounded around y = 0, the electrostatic potential for large values of y, can be written as

@(x,y)

-

Ay/(2eoL)

+

constant, ( y >> L) (7) where A is the totdl net charge inside a infinite cell defined by:

O < x < L ; - m < y < + o r ; o < z < l (8) According to eq. 4 it corresponds to

F(p,v) = ( A / ~ I I E ~ ) In p

+

constant (9) in the w-plane. Thus,

In

p is the only divergent cylindrical harmonic in the '%+problem"

corresponding expansion. It arises from the mapping singularity at p = 0. Hence, for such bounded c-systems the I%-problem" bears a "renonndlizing" charge distribution along

p = 0 with linear charge density

A' =

-

A/2 (10)

which acccnmts for the mapping non-validity in this region. It has to be pointed out that for most practical purposes A = 0 because a macroscopic system seldom bears an effective net electrical charge. Nevertheless, if needed, theoretical situations can be andlysed where A' is not zero and expressions for *(x,y), as well as for the electric field for example, are obtained without ambiguities.

Correlating electric fields based on eqs. 1 and 2 is a straightforward procedure that leads to

E~~ = (2np/L) E~~

,

E = (-Z~P/L) E~~

cY (11)

It can be shown that through this mapping also potential energies are domaik-to-domin preserved, in cases where A = 0.Preceding developnents on the proposed mapping show its usefulness in reducing the problem of an infinite amqy of linear distributions to another comprising only a few linear distributions. Many analogies may be accomplished between original and reduced electrostatic problem on the basis of equivalences or identities here established between various fbnctions and quantities in 5 - m e and their analogues in w-space. In what follows some applications are developed.

3

-

TYPICAL ARRAYS AND S U R F ' HDEL

The simplest system to which this method can be applied is a planar array of equally spaced mompolar linear distributions with a distance L between nearest lines. It is sketched in fig. 2 together with its image system inthe w-plane. If a particular charge distribution of the <-plane is located at site (L/2, 0) and its charge density amounts to A, an equal distribution arises at site (P = 1, v =

n)

of the w-plane (cf. eqs. 3) together with a renomallzing distribution whose charge density amounts to -A/2 (cf. eqs. 8 and 10). Solution of the ww-problernl' is straightf orward and subsequentapplication of eqs

.

^{3 therein}

leads to the solution for the "<-problemf1 potential:

where @(0,0) is zero. A similar and even simpler configration is obtained fnxn the mapping of a planar array of dipolar linear distributions. An element of such array is defined in analogy to a point dipole by substitution of the point charge values by linear charge density yalues. Thus, instead of a dipole moment, a linear+dipole moment density, which we shall call p, characterizes the distribution. When displaced Ro from origin it gives rise to the electric potential

el

(6;

6, fiO)

⁼

[6 . ⁽⁶

^-

%.I

^{/ (2n e0}

^{I 6 - 6o12 1 +}

^{constant (13)}

at point

5.

In fig. 3 the mapped systems are shown for an array of distributions bearing

linear dipole moment density

i;

i n <-space and otherwise identical to the previous monopolar array. Notice that i n this c& A @ = 0 and thuS the w-problem configuration is reduced t o a single dipolar linear distribution a t (0 = 1, u = n), pointing inward. Nevertheless its dipole moment density p is not the same as that _{of its} image on ,-plane. This is due to metric differences Wm these planes. Direct calculations shows that, i n general, dipole m n t densities capments are related by

P = (L/2n P ) pWI;

,

P,, = (-L/2n P ) pWp Applicatlon of this result to present situation leads

,X

to

pwu ⁼ 0

,

pWp = - (2n /L) pSY (15)

Conrbining eqs. 13 and 15 i n accordance with the -try of fig. 3.b, the w-problem is solved.

l3rther use of eq. 1 brings up the final result

r Z -

for the electrostatic potential of a I;-plane dipolar configuration. Again o(0,O) was chosen to be zero. Fig. 4 is a sketch f o r equipotentials inside the configuration "untt celln. For generality that plot w a s made f o r the renormalized potential

v(x,y) = ep (x,Y)/v_ = ap ( X , Y ) / ( P , / ~ E ~ L ) (17) Thus being independent of p

.

For applying the above results to a planar array of atoms, each linear chain i n z - d i r e c t i o n k l l be approximated by a continuous linear distribution bearing a charge density and/or a dipole mcfnent density, according to its particular situation. To these quantities an "integral equivalen~e*~ condition w i l l be imposed as follows: they w i l l be assuned t o ^slnn "macroscopically" up to the same amunt as would the actual l a t t i c e on the same region of the array plane under the same external conditions. The problem of so modelling a square l a t t i c e of at-, with parrvneter L and polarizability a when it is e&edded in a constant electric field, E 0 , i s solved elsewhere (Andion and% C a s t i h , to be published). The dipole moment density p of a single distribution in the model is pioved

to be linearly dependent on the applied field:

6,

= a~ eff

E

⁰

where the "effective linear polarizability density"

= a /&[I

+

L 3 1 f , (19)

with '% eff A

n l r r

K =

%

⁽ⁱ²

+

j 2 ) * I L = 9.0 (20)

being an l~structure factor" /3/ for square lattices. Here represents the surmation on i and j, both ranging over the whole set of natural ru&ers. Besides polarizing the atomic quare l a t t i c e an external field may induce net electric charge on it. Tkis is a possible case of the t i p surface on the Field Ion Microscope (FIM). A s pointed out by Forbes /5/, frcm the total constant primary field, F, only half acts for surface l a t t i c e polarization, the other half being prwided above the metal surface by the same net surface charge alluded to before. !Thus, each distribution frcm the arw &el shall bear both a linear charge density A and a linear m n t density p

.

They rmst obey the "integral equivalencew

conditiodstated earlier for their values. Tkis array is complemented (see, e. g., Forbes /5/) by a plane surface charge distribution located a t y >> L which provides for that polarizing half part of the primary field refered earlier. Thus the electric potential a t a given point the sun

Q(x,Y) =OX (x,Y) + %(x,Y)

+

Q,(y) (21)

of three contributions: that of t h e 1 distribution array, that of the dipolar distribution army and that of the **oppositew surface distributicn, respectively. For the f i r s t of these, A nust be such as t o giye rise, at distant points, to half part of the primary field F. A s can be easily verified this is accomplished for i f A = e L F. The corresponding electric potential i n eq. 21,6 (x,y), is then obtained f r u m eq.

18.

C m b i n i n g eqs. 18 and 19 and taking into account &at E corresponds i n the present case to F/2, the induced dipole mrment on each array model distrigution is obtained:

P, = (aA F)/ [2L(1 + K aA/4n eo LS)

1

(22)

Application of this result into eq. 16 brings the final expression f o r the dipolar distribution array potential, ^Qp (x,y)

.

Finally. the opposite surface distribution contributes with

oT(y) =

-

Fy/2

+

constant (23)

to the total potent;& (eq. 2f), consistently w i t h its f i e l d value: F/2. Fmm these results the electric f i e l d ~ ( x , y ) = -V Q(x,y) is imnediately obtained. In particular its y-c-nt

E Y ( x , ~ ) = E AY ( x , ~ ) + E PY( x , ~ ) + F/2 (24)

where

E = ( ~ / 2 ) [sh(2n y / ~ ) 1 /[cos(2nx/~)

+

ch (2ny/~)1

AY (25)

is the contribution from the A-distribution array and

E =

-

6F[1

+

cos(2nx/L) ch (2ny/~)]/[cos ( 2 n d ~ )

+

ch(2ny/L)

I'

PY ⁽²⁶⁾

with

B = (n aA)/2 €,L3 (1

+

K aA/4n c0 L') (27)

that of the dipolar distribution array. A s a metal surface model the total array shows consistence with the field value inside the conductor. A s a matter of fact i n eq. 24 E(x,y) tends rapidily t o zero as y assumes negative values, i. e., wherever the observation point dips below the metal surface. It can be particularly be shown by direct calculation by eq. 24 that for points below the 010 tungsten surface of %FTM t i p and distant L frcsn that surface, the field magnitude obtained is less than 5.0 x 10 F. This means that the model

a c c q l i s h e s conductor body shielding from the primary field F, without the need to consider inner crystal planes. On the other hand, use of this model for analysing the electric field above that FlM t i p with heliun as imaging gas contributes with some results t o an old dispute i n the area: that of the contribution of surface monopoles and dipoles to the field on an adsorbed h e l i m atom and its p q u e f ! s e s 2 t o the adsorption i t s e l f . Assumption of a tungsten polarizability ow = 1.86 x 10 J. V .m i n place of aA in eq. 27 leads to BW (010) = 1.813.

I f a heliun atom is adsorbed just above a tungsten lattlce atom, the distance between

tBir

centers is the sum r = rHe

+

ry of their redii. This implies y = r = 2.59

61

⁼ 2.59 x 10- m besides x = L/2 f o r the hellurn atom center coordinates. Substitution of these values into eqs. 25 and 26 gives EX = 0.5058 F and E = 0.0213 F for monopolar and dipolar contribution., respectively. I f a vvfieYd anomaly coeffi&8ntvv is defined as (E

-

F)/F, which tends t o zero as y tends to infinity, it can be seen that the monopolar contAbution t o the field ammaly is, i n this cast?, around 0.6 % while that from dipolar f i e l d is around 2.1 %. This result agrees .with Forbes /5/ claim that the binding energy frcm monopole-dipole interaction between tungsten and heliun atoms is i n this case comparable to that from their dipole-dipole

interaction. O n the other hand, the total field value here obtained does not agree with Tsong's /4/ whose "enhancement factorvv fA = (EdF)' amounts here to 1.055 whereas his own value reaches 2-40: The difference may be due to his overestimation of the polarizing electric f i e l d acting upon tungsten atoms. A s a matter of fact there the assl~ned polarizing field corresponds to the total primary field, F, instead of the correct value F/2. The disparity cannot be assigned t o model differences because a t points over a l a t t i c e atom and such a distance apart from it: w = 2.59

61

⁼ 0.820 L athe Ifintegral equivalencevv condition assures a very good approximation between t h i s and his square l a t t i c e model. Another use of this method concerns to the contrast of the

FIM

images, calculated in accordance to Homeier and Kingham's /8/ approach. For doing so, the q l e surface is assumed Lo be an equipotential with the same value as the metal bulk (y + a), the c r i t i c a l surface being then defined i n the usual way /8/. The ratio between the ionization constants a t c r i t i c a l points, as defined by Homeier and Kh&m /8/, can be calculated using Haydock and Kingham's /9/ procedure.

These points correspond, in this case, t o x = 0 and x = L/2 on the c r i t i c a l surface. Details of the calculations w i l l be presented elsewhere (Andion and de Castilho

-

to be published).

Although the proposed mapping w a s here applied to aPoissonvs equation problem, it and its inverse, may dLso be usemi n boundary conditions problems.

This approach, which w a s here applied to a FIM t i p surface, shows the feasibility of analytical models for treating crystals surfaces and other planar atomic arrays. Besides reducing the infinite original periodic configuration t o a f i n i t e and simple one, the mapping preserves essential magnitudes and h c t i o n s , among them the c e l l potential energy and electric charges. F l r t k m r e other, non-preserved quantities are easily correlated by analytical results here obtained, e. g., for the electric field. It also allows for calculations of FIM image contrast parameters. The model limitation of treating one of the surface coordinates as ignorable is counterbalanced by the fact of being totally analytical.

The "integral equivalence" condition allows, in many cases, for a consistent representation of plane arrays of discrete point charges. The whole mathematical vlapparatus" indicates that W t h e r developments of this model are feasible i n which more detailed and accurate surface phenanena analysis may be performed, leading to navel results i n areas as field adsorption and others. Extension t o a more realistic model taking into account periodicity in both surface directions may also be worthwhile.

/1/ Inkson, J. C., J. Phys. C: Solid State Phys.,

10

(1977) 567.

/2/ Gies, P. and Gerhardts, R. R., Phys. Rev. B 33 (1986) 982.

-

/3/ Tsong, T. T. Surface Sci.

2

(1978) 211.

/4/ Forbes, R. G., Surface Sci.

78

(1978) L 504.

/5/ Forbes, R. G. and Wafi, M. L., Surface Sci. 93 (1980) 192.

/6/ Churchill, R. V., in llCaplex Variables and Zplications", M c G m - H i l l , New York (1960).

/7/ Kinghan, D. R., Homeier, H. H. H. and de Castilho, C. M. C., Surface Sci. 152/153 (1985) 55.

/8/ Homeier, H. H. H. and Kk&am, D. R., J. Phys. D: Appl. Phys.

16

(1983) L 115.

/9/ Haydock, R. and Kin&am, D. R., Surface Sci. 103 (1981) 239.

-

Y

Figure 1 - Representation of the mapping given by w = exp (2nic/L) showing (a) the c-plane (c=x+iy) and ( b ) the w-plane ( w = P exp (iv)). The X-axis is mapped onto the circle p = 1 and the Y-axis onto line v = 0. Each region ( l i n e ) i n ( a ) is transformed into the region ( l i n e ) of same texture, in (b)

.

Figure 2 - ( a ) a planar array of lines of charge density X and spacing L located on x=[m(1/2)]L, (N = 0, 1, 2,

.

^..), is mapped onto (b) a single line distribution with the same charge density A on ( p = 1, v = II), plus a

"

r e n o d i z i n g

"

distribution along r, = 0, bearing charge density A ' = - A/2.

Figure 3

-

^{(a) An} a r T of distributions bearing dipole mnent density p with same spacing and

c '

locations as that of fig. 2 is mapped onto + a single dipolar distribution with density

pointing t o the origin n. !Chere i s m

##A1

distributiol i n this caset since ^A = 0. Note that p stq f o r p and p' f o r p

5 w'

Figure 4

-

Sketch showing equipotentials of the dipolar array in fig. 3, over the cell: 0 < x <L.

For generality l i n e s were plotted f o r function v(x,y) =Q ( X , Y ) / ( P ~ / ~ E ~ L ) , not depending of p (see eq. '17). Equpotentials v(x,y) =

+

1

&

asynptotic t o both straight lines x = L/4 and x = 3L/4 and correspond to potentials in the limits y +

-

^{and y +}

^-.